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Theorem List for Metamath Proof Explorer - 27801-27900   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremstoweidlem41 27801* This lemma is used to prove that there exists x as in Lemma 1 of [BrosowskiDeutsh] p. 90: 0 <= x(t) <= 1 for all t in T, x(t) < epsilon for all t in V, x(t) > 1 - epsilon for all t in T \ U. Here we prove the very last step of the proof of Lemma 1: "The result follows from taking x = 1 - qn";. Here is used to represent ε in the paper, and to represent qn in the paper. (Contributed by Glauco Siliprandi, 20-Apr-2017.)

Theoremstoweidlem42 27802* This lemma is used to prove that built as in Lemma 2 of [BrosowskiDeutsh] p. 91, is such that x > 1 - ε on B. Here is used to represent in the paper, and E is used to represent ε in the paper. (Contributed by Glauco Siliprandi, 20-Apr-2017.)

Theoremstoweidlem43 27803* This lemma is used to prove the existence of a function pt as in Lemma 1 of [BrosowskiDeutsh] p. 90 (at the beginning of Lemma 1): for all t in T - U, there exists a function pt in the subalgebra, such that pt( t0 ) = 0 , pt ( t ) > 0, and 0 <= pt <= 1. Hera Z is used for t0 , S is used for t e. T - U , h is used for pt. (Contributed by Glauco Siliprandi, 20-Apr-2017.)

Theoremstoweidlem44 27804* This lemma is used to prove the existence of a function p as in Lemma 1 of [BrosowskiDeutsh] p. 90: p is in the subalgebra, such that 0 <= p <= 1, p(t_0) = 0, and p > 0 on T - U. Z is used to represent t0 in the paper. (Contributed by Glauco Siliprandi, 20-Apr-2017.)

Theoremstoweidlem45 27805* This lemma proves that, given an appropriate (in another theorem we prove such a exists), there exists a function qn as in the proof of Lemma 1 in [BrosowskiDeutsh] p. 91 ( at the top of page 91): 0 <= qn <= 1 , qn < ε on T \ U, and qn > 1 - ε on . We use y to represent the final qn in the paper (the one with n large enough), to represent in the paper, to represent , to represent δ, to represent ε, and to represent . (Contributed by Glauco Siliprandi, 20-Apr-2017.)

Theoremstoweidlem46 27806* This lemma proves that sets U(t) as defined in Lemma 1 of [BrosowskiDeutsh] p. 90, are a cover of T \ U. Using this lemma, in a later theorem we will prove that a finite subcover exists. (Contributed by Glauco Siliprandi, 20-Apr-2017.)

Theoremstoweidlem47 27807* Subtracting a constant from a real continuous function gives another continuous function. (Contributed by Glauco Siliprandi, 20-Apr-2017.)

Theoremstoweidlem48 27808* This lemma is used to prove that built as in Lemma 2 of [BrosowskiDeutsh] p. 91, is such that x < ε on . Here is used to represent in the paper, is used to represent ε in the paper, and is used to represent in the paper (because is always used to represent the subalgebra). (Contributed by Glauco Siliprandi, 20-Apr-2017.)

Theoremstoweidlem49 27809* There exists a function qn as in the proof of Lemma 1 in [BrosowskiDeutsh] p. 91 (at the top of page 91): 0 <= qn <= 1 , qn < ε on , and qn > 1 - ε on . Here y is used to represent the final qn in the paper (the one with n large enough), represents in the paper, represents , represents δ, represents ε, and represents . (Contributed by Glauco Siliprandi, 20-Apr-2017.)

Theoremstoweidlem50 27810* This lemma proves that sets U(t) as defined in Lemma 1 of [BrosowskiDeutsh] p. 90, contain a finite subcover of T \ U. (Contributed by Glauco Siliprandi, 20-Apr-2017.)

Theoremstoweidlem51 27811* There exists a function x as in the proof of Lemma 2 in [BrosowskiDeutsh] p. 91. Here is used to represent in the paper, because here is used for the subalgebra of functions. is used to represent ε in the paper. (Contributed by Glauco Siliprandi, 20-Apr-2017.)

Theoremstoweidlem52 27812* There exists a neighborood V as in Lemma 1 of [BrosowskiDeutsh] p. 90. Here Z is used to represent t0 in the paper, and v is used to represent V in the paper. (Contributed by Glauco Siliprandi, 20-Apr-2017.)

Theoremstoweidlem53 27813* This lemma is used to prove the existence of a function p as in Lemma 1 of [BrosowskiDeutsh] p. 90: p is in the subalgebra, such that 0 <= p <= 1, p(t_0) = 0, and p > 0 on T - U. (Contributed by Glauco Siliprandi, 20-Apr-2017.)

Theoremstoweidlem54 27814* There exists a function as in the proof of Lemma 2 in [BrosowskiDeutsh] p. 91. Here is used to represent in the paper, because here is used for the subalgebra of functions. is used to represent ε in the paper. (Contributed by Glauco Siliprandi, 20-Apr-2017.)

Theoremstoweidlem55 27815* This lemma proves the existence of a function p as in the proof of Lemma 1 in [BrosowskiDeutsh] p. 90: p is in the subalgebra, such that 0 <= p <= 1, p(t_0) = 0, and p > 0 on T - U. Here Z is used to represent t0 in the paper. (Contributed by Glauco Siliprandi, 20-Apr-2017.)

Theoremstoweidlem56 27816* This theorem proves Lemma 1 in [BrosowskiDeutsh] p. 90. Here is used to represent t0 in the paper, is used to represent in the paper, and is used to represent ε (Contributed by Glauco Siliprandi, 20-Apr-2017.)

Theoremstoweidlem57 27817* There exists a function x as in the proof of Lemma 2 in [BrosowskiDeutsh] p. 91. In this theorem, it is proven the non-trivial case (the closed set D is nonempty). Here D is used to represent A in the paper, because the variable A is used for the subalgebra of functions. (Contributed by Glauco Siliprandi, 20-Apr-2017.)

Theoremstoweidlem58 27818* This theorem proves Lemma 2 in [BrosowskiDeutsh] p. 91. Here D is used to represent the set A of Lemma 2, because here the variable A is used for the subalgebra of functions. (Contributed by Glauco Siliprandi, 20-Apr-2017.)

Theoremstoweidlem59 27819* This lemma proves that there exists a function as in the proof in [BrosowskiDeutsh] p. 91, after Lemma 2: xj is in the subalgebra, 0 <= xj <= 1, xj < ε / n on Aj (meaning A in the paper), xj > 1 - \epslon / n on Bj. Here is used to represent A in the paper (because A is used for the subalgebra of functions), is used to represent ε. (Contributed by Glauco Siliprandi, 20-Apr-2017.)

Theoremstoweidlem60 27820* This lemma proves that there exists a function g as in the proof in [BrosowskiDeutsh] p. 91 (this parte of the proof actually spans through pages 91-92): g is in the subalgebra, and for all in , there is a such that (j-4/3)*ε < f(t) <= (j-1/3)*ε and (j-4/3)*ε < g(t) < (j+1/3)*ε. Here is used to represent f in the paper, and is used to represent ε. (Contributed by Glauco Siliprandi, 20-Apr-2017.)

Theoremstoweidlem61 27821* This lemma proves that there exists a function as in the proof in [BrosowskiDeutsh] p. 92: is in the subalgebra, and for all in , abs( f(t) - g(t) ) < 2*ε. Here is used to represent f in the paper, and is used to represent ε. For this lemma there's the further assumption that the function to be approximated is nonnegative (this assumption is removed in a later theorem). (Contributed by Glauco Siliprandi, 20-Apr-2017.)

Theoremstoweidlem62 27822* This theorem proves the Stone Weierstrass theorem for the non-trivial case in which T is nonempty. The proof follows [BrosowskiDeutsh] p. 89 (through page 92). (Contributed by Glauco Siliprandi, 20-Apr-2017.)

Theoremstoweid 27823* This theorem proves the Stone-Weierstrass theorem for real valued functions: let be a compact topology on , and be the set of real continuous functions on . Assume that is a subalgebra of (closed under addition and multiplication of functions) containing constant functions and discriminating points (if and are distinct points in , then there exists a function in such that h(r) is distinct from h(t) ). Then, for any continuous function and for any positive real , there exists a function in the subalgebra , such that approximates up to ( represents the usual ε value). As a classical example, given any a,b reals, the closed interval could be taken, along with the subalgebra of real polynomials on , and then use this theorem to easily prove that real polynomials are dense in the standard metric space of continuous functions on . The proof and lemmas are written following [BrosowskiDeutsh] p. 89 (through page 92). Some effort is put in avoiding the use of the axiom of choice. (Contributed by Glauco Siliprandi, 20-Apr-2017.)

Theoremstowei 27824* This theorem proves the Stone-Weierstrass theorem for real valued functions: let be a compact topology on , and be the set of real continuous functions on . Assume that is a subalgebra of (closed under addition and multiplication of functions) containing constant functions and discriminating points (if and are distinct points in , then there exists a function in such that h(r) is distinct from h(t) ). Then, for any continuous function and for any positive real , there exists a function in the subalgebra , such that approximates up to ( represents the usual ε value). As a classical example, given any a,b reals, the closed interval could be taken, along with the subalgebra of real polynomials on , and then use this theorem to easily prove that real polynomials are dense in the standard metric space of continuous functions on . The proof and lemmas are written following [BrosowskiDeutsh] p. 89 (through page 92). Some effort is put in avoiding the use of the axiom of choice. The deduction version of this theorem is stoweid 27823: often times it will be better to use stoweid 27823 in other proofs (but this version is probably easier to be read and understood). (Contributed by Glauco Siliprandi, 20-Apr-2017.)

18.20.7  Wallis' product for π

Theoremwallispilem1 27825* is monotone: increasing the exponent, the integral decreases. (Contributed by Glauco Siliprandi, 29-Jun-2017.)

Theoremwallispilem2 27826* A first set of properties for the sequence that will be used in the proof of the Wallis product formula (Contributed by Glauco Siliprandi, 29-Jun-2017.)

Theoremwallispilem3 27827* I maps to real values (Contributed by Glauco Siliprandi, 29-Jun-2017.)

Theoremwallispilem4 27828* maps to explicit expression for the ratio of two consecutive values of (Contributed by Glauco Siliprandi, 30-Jun-2017.)

Theoremwallispilem5 27829* The sequence converges to 1. (Contributed by Glauco Siliprandi, 30-Jun-2017.)

Theoremwallispi 27830* Wallis' formula for π : Wallis' product converges to π / 2 . (Contributed by Glauco Siliprandi, 29-Jun-2017.)

Theoremwallispi2lem1 27831 An intermediate step between the first version of the Wallis' formula for π and the second version of Wallis' formula. This second version will then be used to prove Stirling's approximation formula for the factorial. (Contributed by Glauco Siliprandi, 30-Jun-2017.)

Theoremwallispi2lem2 27832 Two expressions are proven to be equal, and this is used to complete the proof of the second version of Wallis' formula for π . (Contributed by Glauco Siliprandi, 30-Jun-2017.)

Theoremwallispi2 27833 An alternative version of Wallis' formula for π ; this second formula uses factorials and it is later used to proof Stirling's approximation formula. (Contributed by Glauco Siliprandi, 29-Jun-2017.)

18.20.8  Stirling's approximation formula for ` n ` factorial

Theoremstirlinglem1 27834 A simple limit of fractions is computed. (Contributed by Glauco Siliprandi, 30-Jun-2017.)

Theoremstirlinglem2 27835 maps to positive reals. (Contributed by Glauco Siliprandi, 29-Jun-2017.)

Theoremstirlinglem3 27836 Long but simple algebraic transformations are applied to show that , the Wallis formula for π , can be expressed in terms of , the Stirling's approximation formula for the factorial, up to a constant factor. This will allow (in a later theorem) to determine the right constant factor to be put into the , in order to get the exact Stirling's formula. (Contributed by Glauco Siliprandi, 29-Jun-2017.)

Theoremstirlinglem4 27837* Algebraic manipulation of n ) - ( B . It will be used in other theorems to show that is decreasing. (Contributed by Glauco Siliprandi, 29-Jun-2017.)

Theoremstirlinglem5 27838* If is between and , then a series (without alternating negative and positive terms) is given that converges to log (1+T)/(1-T) . (Contributed by Glauco Siliprandi, 29-Jun-2017.)

Theoremstirlinglem6 27839* A series that converges to log (N+1)/N (Contributed by Glauco Siliprandi, 29-Jun-2017.)

Theoremstirlinglem7 27840* Algebraic manipulation of the formula for J(n) (Contributed by Glauco Siliprandi, 29-Jun-2017.)

Theoremstirlinglem8 27841 If converges to , then converges to C^2 . (Contributed by Glauco Siliprandi, 29-Jun-2017.)

Theoremstirlinglem9 27842* is expressed as a limit of a series. This result will be used both to prove that is decreasing and to prove that is bounded (below). It will follow that converges in the reals. (Contributed by Glauco Siliprandi, 29-Jun-2017.)

Theoremstirlinglem10 27843* A bound for any B(N)-B(N + 1) that will allow to find a lower bound for the whole sequence. (Contributed by Glauco Siliprandi, 29-Jun-2017.)

Theoremstirlinglem11 27844* is decreasing. (Contributed by Glauco Siliprandi, 29-Jun-2017.)

Theoremstirlinglem12 27845* The sequence is bounded below. (Contributed by Glauco Siliprandi, 29-Jun-2017.)

Theoremstirlinglem13 27846* is decreasing and has a lower bound, then it converges. Since is , in another theorem it is proven that converges as well. (Contributed by Glauco Siliprandi, 29-Jun-2017.)

Theoremstirlinglem14 27847* The sequence converges to a positive real. This proves that the Stirling's formula converges to the factorial, up to a constant. In another theorem, using Wallis' formula for π& , such constant is exactly determined, thus proving the Stirling's formula. (Contributed by Glauco Siliprandi, 29-Jun-2017.)

Theoremstirlinglem15 27848* The Stirling's formula is proven using a number of local definitions. The main theorem stirling 27849 will use this final lemma, but it will not expose the local definitions. (Contributed by Glauco Siliprandi, 29-Jun-2017.)

Theoremstirling 27849 Stirling's approximation formula for factorial. The proof follows two major steps: first it is proven that and factorial are asymptotically equivalent, up to an unknown constant. Then, using Wallis' formula for π it is proven that the unknown constant is the square root of π and then the exact Stirling's formula is established. (Contributed by Glauco Siliprandi, 29-Jun-2017.)

Theoremstirlingr 27850 Stirling's approximation formula for factorial: here convergence is expressed with respect to the standard topology on the reals. The main theorem stirling 27849 is proven for convergence in the topology of complex numbers. The variable is used to denote convergence with respect to the standard topology on the reals. (Contributed by Glauco Siliprandi, 29-Jun-2017.)

18.21  Mathbox for Saveliy Skresanov

18.21.1  Ceva's theorem

Theoremsigarval 27851* Define the signed area by treating complex numbers as vectors with two components. (Contributed by Saveliy Skresanov, 19-Sep-2017.)

Theoremsigarim 27852* Signed area takes value in reals. (Contributed by Saveliy Skresanov, 19-Sep-2017.)

Theoremsigarac 27853* Signed area is anticommutative. (Contributed by Saveliy Skresanov, 19-Sep-2017.)

Theoremsigaraf 27854* Signed area is additive by the first argument. (Contributed by Saveliy Skresanov, 19-Sep-2017.)

Theoremsigarmf 27855* Signed area is additive (with respect to subtraction) by the first argument. (Contributed by Saveliy Skresanov, 19-Sep-2017.)

Theoremsigaras 27856* Signed area is additive by the second argument. (Contributed by Saveliy Skresanov, 19-Sep-2017.)

Theoremsigarms 27857* Signed area is additive (with respect to subtraction) by the second argument. (Contributed by Saveliy Skresanov, 19-Sep-2017.)

Theoremsigarls 27858* Signed area is linear by the second argument. (Contributed by Saveliy Skresanov, 19-Sep-2017.)

Theoremsigarid 27859* Signed area of a flat parallelogram is zero. (Contributed by Saveliy Skresanov, 20-Sep-2017.)

Theoremsigarexp 27860* Expand the signed area formula by linearity. (Contributed by Saveliy Skresanov, 20-Sep-2017.)

Theoremsigarperm 27861* Signed area acts as a double area of a triangle . Here we prove that cyclically permuting the vertices doesn't change the area. (Contributed by Saveliy Skresanov, 20-Sep-2017.)

Theoremsigardiv 27862* If signed area between vectors and is zero, then those vectors lie on the same line. (Contributed by Saveliy Skresanov, 22-Sep-2017.)

Theoremsigarimcd 27863* Signed area takes value in complex numbers. Deduction version. (Contributed by Saveliy Skresanov, 23-Sep-2017.)

Theoremsigariz 27864* If signed area is zero, the signed area with swapped arguments is also zero. Deduction version. ( Contributed by Saveliy Skresanov, 23-Sep-2017.) (Contributed by Saveliy Skresanov, 24-Sep-2017.)

Theoremsigarcol 27865* Given three points , and such that , the point lies on the line going through and iff the corresponding signed area is zero. That justifies the usage of signed area as a collinearity indicator. (Contributed by Saveliy Skresanov, 22-Sep-2017.)

Theoremsharhght 27866* Let be a triangle, and let lie on the line . Then (doubled) areas of triangles and relate as lengths of corresponding bases and . (Contributed by Saveliy Skresanov, 23-Sep-2017.)

Theoremsigaradd 27867* Subtracting (double) area of from yields the (double) area of . (Contributed by Saveliy Skresanov, 23-Sep-2017.)

Theoremcevathlem1 27868 Ceva's theorem first lemma. Multiplies three identities and divides by the common factors. (Contributed by Saveliy Skresanov, 24-Sep-2017.)

Theoremcevathlem2 27869* Ceva's theorem second lemma. Relate (doubled) areas of triangles and with of segments and . (Contributed by Saveliy Skresanov, 24-Sep-2017.)

Theoremcevath 27870* Ceva's theorem. Let be a triangle and let points , and lie on sides , , correspondingly. Suppose that cevians , and intersect at one point . Then triangle's sides are partitioned into segments and their lengths satisfy a certain identity. Here we obtain a bit stronger version by using complex numbers themselves instead of their absolute values.

The proof goes by applying cevathlem2 27869 three times and then using cevathlem1 27868 to multiply obtained identities and prove the theorem.

In the theorem statement we are using function as a collinearity indicator. For justification of that use, see sigarcol 27865. (Contributed by Saveliy Skresanov, 24-Sep-2017.)

18.22  Mathbox for Jarvin Udandy

TheoremhirstL-ax3 27871 The third axiom of a system called "L" but proven to be a theorem since set.mm uses a different third axiom. This is named hirst after Holly P. Hirst and Jeffry L. Hirst. Axiom A3 of [Mendelson] p. 35. (Contributed by Jarvin Udandy, 7-Feb-2015.) (Proof modification is discouraged.)

Theoremax3h 27872 Recovery of ax-3 7 from hirstL-ax3 27871. (Contributed by Jarvin Udandy, 3-Jul-2015.)

Theoremaibandbiaiffaiffb 27873 A closed form showing (a implies b and b implies a) same-as (a same-as b) (Contributed by Jarvin Udandy, 3-Sep-2016.)

Theoremaibandbiaiaiffb 27874 A closed form showing (a implies b and b implies a) implies (a same-as b) (Contributed by Jarvin Udandy, 3-Sep-2016.)

Theoremnotatnand 27875 Do not use. Use intnanr instead. Given not a, there exists a proof for not (a and b). (Contributed by Jarvin Udandy, 31-Aug-2016.)

Theoremaistia 27876 Given a is equivalent to T., there exists a proof for a. (Contributed by Jarvin Udandy, 30-Aug-2016.)

Theoremaisfina 27877 Given a is equivalent to F., there exists a proof for not a. (Contributed by Jarvin Udandy, 30-Aug-2016.)

Theorembothtbothsame 27878 Given both a,b are equivalent to T., there exists a proof for a is the same as b. (Contributed by Jarvin Udandy, 31-Aug-2016.)

Theorembothfbothsame 27879 Given both a,b are equivalent to F., there exists a proof for a is the same as b. (Contributed by Jarvin Udandy, 31-Aug-2016.)

Theoremaiffbbtat 27880 Given a is equivalent to b, b is equivalent to T. there exists a proof for a is equivalent to T. (Contributed by Jarvin Udandy, 29-Aug-2016.)

Theoremaisbbisfaisf 27881 Given a is equivalent to b, b is equivalent to F. there exists a proof for a is equivalent to F. (Contributed by Jarvin Udandy, 30-Aug-2016.)

Theoremaxorbtnotaiffb 27882 Given a is exclusive to b, there exists a proof for (not (a if-and-only-if b)) df-xor is a closed form of this. (Contributed by Jarvin Udandy, 7-Sep-2016.)

Theoremaiffnbandciffatnotciffb 27883 Given a is equivalent to NOT b, c is equivalent to a. there exists a proof for ( not ( c iff b ) ). (Contributed by Jarvin Udandy, 7-Sep-2016.)

Theoremaxorbciffatcxorb 27884 Given a is equivalent to NOT b, c is equivalent to a. there exists a proof for ( c xor b ) . (Contributed by Jarvin Udandy, 7-Sep-2016.)

Theoremaibnbna 27885 Given a implies b, not b, there exists a proof for not a. (Contributed by Jarvin Udandy, 1-Sep-2016.)

Theoremaibnbaif 27886 Given a implies b, not b, there exists a proof for a is F. (Contributed by Jarvin Udandy, 1-Sep-2016.)

Theoremaiffbtbat 27887 Given a is equivalent to b, T. is equivalent to b. there exists a proof for a is equivalent to T. (Contributed by Jarvin Udandy, 29-Aug-2016.)

Theoremastbstanbst 27888 Given a is equivalent to T., also given that b is equivalent to T, there exists a proof for a and b is equivalent to T. (Contributed by Jarvin Udandy, 29-Aug-2016.)

Theoremaistbistaandb 27889 Given a is equivalent to T., also given that b is equivalent to T, there exists a proof for (a and b). (Contributed by Jarvin Udandy, 9-Sep-2016.)

Theoremaisbnaxb 27890 Given a is equivalent to b, there exists a proof for (not (a xor b)). (Contributed by Jarvin Udandy, 28-Aug-2016.)

Theoremiatbtatnnb 27891 Given a implies b, there exists a proof for a implies not not b. (Contributed by Jarvin Udandy, 2-Sep-2016.)

Theorematbiffatnnb 27892 If a implies b, then a implies not not b (Contributed by Jarvin Udandy, 28-Aug-2016.)

Theorembisaiaisb 27893 Application of bicom1 with a, b swapped. (Contributed by Jarvin Udandy, 31-Aug-2016.)

Theorematbiffatnnbalt 27894 If a implies b, then a implies not not b (Contributed by Jarvin Udandy, 29-Aug-2016.)

Theoremabnotbtaxb 27895 Assuming a, not b, there exists a proof a-xor-b.) (Contributed by Jarvin Udandy, 31-Aug-2016.)

Theoremabnotataxb 27896 Assuming not a, b, there exists a proof a-xor-b.) (Contributed by Jarvin Udandy, 31-Aug-2016.)

Theoremconimpf 27897 Assuming a, not b, and a implies b, there exists a proof that a is false.) (Contributed by Jarvin Udandy, 28-Aug-2016.)

Theoremconimpfalt 27898 Assuming a, not b, and a implies b, there exists a proof that a is false.) (Contributed by Jarvin Udandy, 29-Aug-2016.)

Theoremaistbisfiaxb 27899 Given a is equivalent to T., Given b is equivalent to F. there exists a proof for a-xor-b. (Contributed by Jarvin Udandy, 31-Aug-2016.)

Theoremaisfbistiaxb 27900 Given a is equivalent to F., Given b is equivalent to T., there exists a proof for a-xor-b. (Contributed by Jarvin Udandy, 31-Aug-2016.)

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